User:Richard Oberrieder/sandbox

Paradox of the latest black hole

Abstract :	 In a very distant future, the mass of the final black hole will not be a measurable parameter. This information will disappear without assumption of Hawking radiation and black hole evaporation. Only a quantification of space-time continuum avoids this paradox.

I - Introduction

Assume the existence of an observer in the distant future after a long period of accelerating expansion of the universe [1]. All celestial bodies are far from each other so as to exceed the cosmological horizon of the observer except two Schwarzschild Black Holes, electrically neutral and without rotation [2]. By definition, for the Schwarzschild black holes : mass M strictly positive: M> 0; that the electric charge Q is zero: Q = 0; which is zero angular momentum J: J = 0 (no axial rotation); whose gravitational singularity is a point; whose event horizon is a hypersurface of radius equal to the Schwarzschild radius. The visible universe would be summarized to the observer O, two Black Holes BH1 and BH2 and cosmic microwave background on the cosmological horizon. In this hypothesis, we neglect the possible Hawking radiation of black holes [3].

II - Measures the radius of black holes by the observer

Even if a black hole does not emit any radiation, the observer O can always measure geometrically its radius R because its disk is silhouetted against the cosmic microwave background with shadow and Einstein rings [4]. But No hair theorem [5] implies that a black hole, electrically neutral and without rotation, is fully characterized by its mass M. The black hole has no detail, no 'hair' visible to its own horizon. Its radius R determines his only observable parameter M through the expression of the Schwarzschild radius : R= 2GM/c^2 	and	M= (c^2 R)/2G The observer O can measure the mass M of each black hole by measuring its radius R. Measurements of radii R1 and R2, the observer O deduce the masses M1 of BH1 and M2 of BH2 because the mass M quantifies the radius R of the black hole.

III - If the observer is a singularity

Now, suppose that the observer O is not a third object, but the singularity S1 of BH1. This singularity S1 sees the shadow disk of the other black hole BH2, measures its radius R2 and therefore its mass M2. Similarly, the singularity S2 sees the shadow disk of BH1, measures its radius R1 and therefore its mass M1. In this remote and very simplified universe, the mass of black holes remains an observable parameter. Nothing can escape the respective horizon of each black hole : the singularities S1 and S2 can’t share information on their respective measures. S1 measures the mass M2 but can’t communicate it to S2. Similarly, S2 measures the mass M1 but can’t communicate it to S1.

IV - Final black hole paradox

Suppose time passes again until the two black holes merge or until they move away so much from each other that they exceed their respective cosmological horizon. In the observable universe by the final singularity Sf, there would be only the final black hole BHf and the cosmic microwave background. The Sf singularity observes a cosmic background without shadow and no Einstein ring. If the singularity Sf can’t observe the radius of BHf, its proper mass Mf is not an observable parameter! The paradox is : with two black holes, there are measurable information "black hole mass" but, for the final black hole, the measurable information disappears with no evaporation of the black hole.

V - Where is the measurable information "black hole mass"?

To understand the paradox, let us return to the case on two black holes. If each singularity can measure the mass of the black hole but not its proper mass, two developments are to be seen. If the two black holes are moving away to exceed their respective cosmological horizon, we can consider that the measurable information "black hole mass" leaves the observable universe. This information disappears to the cosmological horizon of the observable universe. But if the two black holes merge, there are one singularity Sf which can’t measure its proper mass Mf, addition of masses M1 and M2. The measurable information "black hole mass" disappears in the observable universe of the final singularity Sf! We could say that, by combining both, S1 and S2 singularities communicate information to the M1 and M2 measures. But then Sf has Mf information no longer measurable after merge!

VI - How to solve this paradox?

This paradox can be avoided if we consider that the singularity of a black hole can measure its proper mass. The last measurable parameter by the point final singularity Sf is the proper time. Likewise the radius of a black hole is quantified according to its mass by Schwarzschild radius, R= 2GM/c^2 	and	M= (c^2 R)/2G we must consider that the proper time of the singularity S is quantified by the equation : 		ΔT= R/c=2GM/c^3 		and	M= (c^3 ΔT)/2G ΔT is the minimum measurable time by S singularity of a black hole of mass M [6]. The information "black hole mass" remains measurable even for the final black hole of the observable universe. Any singularity S measures its proper mass M measuring its proper time quanta ΔT.

VII - Conclusion

Taking the convention c = G = 1, in a repository (S, x, y, z, ct) related to the point singularity S, the singularity S of a black hole BH has a mass M which quantifies the space-time continuum surrounding as a 4-sphere of center S and radius equal 2M.

[1] Measurements of Omega and Lambda from 42 High-Redshift Supernovae S. PERLMUTTER1, G. ALDERING, G. GOLDHABER1, R.A. KNOP, P. NUGENT, P. G. CASTRO2, S. DEUSTUA, S. FABBRO3, A. GOOBAR4, D. E. GROOM, I. M. HOOK5, A. G. KIM1,6, M. Y. KIM, J. C. LEE7, N. J. NUNES2, R. PAIN3, C. R. PENNYPACKER8, R. QUIMBY fr.arXiv.org > astro-ph > arXiv:astro-ph/9812133v1

[2] Karl Schwarzschild, On the Gravitational Field of a Mass Point according to Einstein’s Theory, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, Phys.-Math.Klasse 1916, 189-196

[3] S. W. Hawking, Particle Creation by Black Holes, Communications in Mathematical Physics, 43, 199-220 (1975) ; Erratum ibid., 46, 206 (1976)

[4] Kochanek, C.S.; C.R. Keeton and B.A. McLeod (2001). "The Importance of Einstein Rings". The Astrophysical Journal 547 (1): 50–59. arXiv:astro-ph/0006116. Bibcode:2001ApJ...547...50K. doi:10.1086/318350.

[5] Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. pp. 875–876. ISBN 0716703343. Retrieved 24 January 2013.

Paradox of the latest black hole
Abstract :	 In a very distant future, the mass of the final black hole will not be a measurable parameter. This information will disappear without assumption of Hawking radiation and black hole evaporation. Only a quantification of space-time continuum avoids this paradox.

I - Introduction

Assume the existence of an observer in the distant future after a long period of accelerating expansion of the universe [1]. All celestial bodies are far from each other so as to exceed the cosmological horizon of the observer except two Schwarzschild Black Holes, electrically neutral and without rotation [2]. By definition, for the Schwarzschild black holes : mass M strictly positive: M> 0; that the electric charge Q is zero: Q = 0; which is zero angular momentum J: J = 0 (no axial rotation); whose gravitational singularity is a point; whose event horizon is a hypersurface of radius equal to the Schwarzschild radius. The visible universe would be summarized to the observer O, two Black Holes BH1 and BH2 and cosmic microwave background on the cosmological horizon. In this hypothesis, we neglect the possible Hawking radiation of black holes [3].

II - Measures the radius of black holes by the observer

Even if a black hole does not emit any radiation, the observer O can always measure geometrically its radius R because its disk is silhouetted against the cosmic microwave background with shadow and Einstein rings [4]. But No hair theorem [5] implies that a black hole, electrically neutral and without rotation, is fully characterized by its mass M. The black hole has no detail, no 'hair' visible to its own horizon. Its radius R determines his only observable parameter M through the expression of the Schwarzschild radius : R= 2GM/c^2 	and	M= (c^2 R)/2G The observer O can measure the mass M of each black hole by measuring its radius R. Measurements of radii R1 and R2, the observer O deduce the masses M1 of BH1 and M2 of BH2 because the mass M quantifies the radius R of the black hole.

III - If the observer is a singularity

Now, suppose that the observer O is not a third object, but the singularity S1 of BH1. This singularity S1 sees the shadow disk of the other black hole BH2, measures its radius R2 and therefore its mass M2. Similarly, the singularity S2 sees the shadow disk of BH1, measures its radius R1 and therefore its mass M1. In this remote and very simplified universe, the mass of black holes remains an observable parameter. Nothing can escape the respective horizon of each black hole : the singularities S1 and S2 can’t share information on their respective measures. S1 measures the mass M2 but can’t communicate it to S2. Similarly, S2 measures the mass M1 but can’t communicate it to S1.

IV - Final black hole paradox

Suppose time passes again until the two black holes merge or until they move away so much from each other that they exceed their respective cosmological horizon. In the observable universe by the final singularity Sf, there would be only the final black hole BHf and the cosmic microwave background. The Sf singularity observes a cosmic background without shadow and no Einstein ring. If the singularity Sf can’t observe the radius of BHf, its proper mass Mf is not an observable parameter! The paradox is : with two black holes, there are measurable information "black hole mass" but, for the final black hole, the measurable information disappears with no evaporation of the black hole.

V - Where is the measurable information "black hole mass"?

To understand the paradox, let us return to the case on two black holes. If each singularity can measure the mass of the black hole but not its proper mass, two developments are to be seen. If the two black holes are moving away to exceed their respective cosmological horizon, we can consider that the measurable information "black hole mass" leaves the observable universe. This information disappears to the cosmological horizon of the observable universe. But if the two black holes merge, there are one singularity Sf which can’t measure its proper mass Mf, addition of masses M1 and M2. The measurable information "black hole mass" disappears in the observable universe of the final singularity Sf! We could say that, by combining both, S1 and S2 singularities communicate information to the M1 and M2 measures. But then Sf has Mf information no longer measurable after merge!

VI - How to solve this paradox?

This paradox can be avoided if we consider that the singularity of a black hole can measure its proper mass. The last measurable parameter by the point final singularity Sf is the proper time. Likewise the radius of a black hole is quantified according to its mass by Schwarzschild radius, R= 2GM/c^2 	and	M= (c^2 R)/2G we must consider that the proper time of the singularity S is quantified by the equation : 		ΔT= R/c=2GM/c^3 		and	M= (c^3 ΔT)/2G ΔT is the minimum measurable time by S singularity of a black hole of mass M [6]. The information "black hole mass" remains measurable even for the final black hole of the observable universe. Any singularity S measures its proper mass M measuring its proper time quanta ΔT.

VII - Conclusion

Taking the convention c = G = 1, in a repository (S, x, y, z, ct) related to the point singularity S, the singularity S of a black hole BH has a mass M which quantifies the space-time continuum surrounding as a 4-sphere of center S and radius equal 2M.

[1] Measurements of Omega and Lambda from 42 High-Redshift Supernovae S. PERLMUTTER1, G. ALDERING, G. GOLDHABER1, R.A. KNOP, P. NUGENT, P. G. CASTRO2, S. DEUSTUA, S. FABBRO3, A. GOOBAR4, D. E. GROOM, I. M. HOOK5, A. G. KIM1,6, M. Y. KIM, J. C. LEE7, N. J. NUNES2, R. PAIN3, C. R. PENNYPACKER8, R. QUIMBY fr.arXiv.org > astro-ph > arXiv:astro-ph/9812133v1

[2] Karl Schwarzschild, On the Gravitational Field of a Mass Point according to Einstein’s Theory, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, Phys.-Math.Klasse 1916, 189-196

[3] S. W. Hawking, Particle Creation by Black Holes, Communications in Mathematical Physics, 43, 199-220 (1975) ; Erratum ibid., 46, 206 (1976)

[4] Kochanek, C.S.; C.R. Keeton and B.A. McLeod (2001). "The Importance of Einstein Rings". The Astrophysical Journal 547 (1): 50–59. arXiv:astro-ph/0006116. Bibcode:2001ApJ...547...50K. doi:10.1086/318350.

[5] Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. pp. 875–876. ISBN 0716703343. Retrieved 24 January 2013.